The Picard group of the moduli space of curves with level structures

Understand

For $4 \nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures.

• In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group (in a previous paper, the author determined this rationally).
• This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author.
• Finally, along the way we calculate the first homology group of the mod $L$ symplectic group with coefficients in the adjoint representation.